The known classes of DHOST theories have the action of the following form [1]}, [2]}, [3]}, [4]}, [5]}:
\(\bar{S}[\bar{g}_{\mu \nu },\phi ]=\int 4x\sqrt{-\bar{g}}\left[ \bar{a}_0(\phi ,\bar{X})\bar{R}+\bar{a}_1(\phi ,\bar{X})\bar{G}^{\mu \nu }\bar{\nabla }_\mu \bar{\nabla }_\nu \phi +\bar{\mathcal {L}}(\bar{g}_{\mu \nu },\phi ,\bar{\nabla }_\mu \phi ,\bar{\nabla }_\mu \bar{\nabla }_\nu \phi ) \right], \) 
